Generally, in computer graphics, objects are represented as surfaces, with the surfaces being represented by meshes. A mesh consists of a set of vertices, or points in three-dimensional space, interconnected by edges. The edges define polygonal faces, which may be in the form of triangles, quadrilaterals, and so forth. In some computer graphic operations, it is desirable to generate a representation of a surface at a finer resolution than a current representation. There are several popular methodologies for generating a representation of a surface at a finer resolution than a current representation, including a Catmull-Clark surface subdivision methodology and a Loop surface subdivision methodology. In other operations, it can also be desirable to generate from a mesh representing a surface at one level of resolution, a mesh at a coarser resolution. A methodology for generating a representation of a surface at a coarser resolution than a current representation, and a methodology for generating such a surface representation, based on the Loop methodology, is described in D. Zorin, et al., “Interactive Multiresolution Mesh Editing, SIGGRAPH 1997, Computer Graphics Proceedings, pp. 259-268, particularly p. 262.
The methodology described in Zorin relies on a method for smoothing triangular meshes that is described in G. Taubin, “A signal processing approach to fair surface design,” SIGGRAPH 1995, Computer Graphics Conference Proceedings, pp. 351-358. Taubin's smoothing methodology consists of two linear filtering steps, the first being a low-pass filter step, followed by a high-pass filter step. The high-pass filter step is used to cancel out a mesh-shrinking effect that is induced by the low-pass filter step. The filtering operations described by Taubin have the problem that, as a consequence of the linearity of the filtering procedure, artifacts referred to as Gibbs ripples are introduced in the smoothed mesh as a result of discontinuities in the input mesh. When used in generating a coarse level mesh, Taubin's filtering produces Gibbs ripples at the places on the surface where the fine level mesh has sharp variations. Although Taubin's smoothing methodology may be configured with the help of parameters, no filter can play the role of an inverse of the Loop subdivision rule. Accordingly, in a multi-resolution representation of a Loop subdivision surface, there will be redundant detail information.